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 in FinanceIgor Hlivka - MUFG Securities International
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Introduction</Font></Text-field>Many aspects of Financial Economics are technical in nature and require mathematical formulation to solve them.Extensive use of mathematical methods in Finance has led to the development of Mathematical Finance - a distinct subgroup of mathematical and statistical rules and theorems that deal with particular problems arising in financial analysis. In pure mathematical terms, problems of Financial Economics are typically reduced to (i) equations solving, (ii) optimization routines, (iii) simulations, (iv) parameters estimation or (v) process modeling which are all well-suited for computer algebra packages using symbolic manipulation and highly efficient numerical solvers.The purpose of this document is to demonstrate how Maple - with its powerful analytical engine and robust numerical routines - can be used to tackle various topics in Financial Economics / Mathematics, and lead to elegant solutions through its symbolic and numerical interfaces.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Interest base conversion, compounding and discounting</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Rates compounding</Font></Text-field>There are many many different rates in the economy - they differ by:Maturity: short-term vs. long-term ratesStructure: zero-coupon (spot), par, forward rates etc.Compounding frequency: annual, semi-annual, quarterly, daily, continuously compounded etc.Here we examine how one can easily obtain conversion formulas for interest rates under various compounding regimes:Assume we are given a rate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== with discrete compounding frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRK1swLDUxLDI1NV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1JI21uR0YkNiVRIjFGJ0Y1L0Y5USdub3JtYWxGJ0Y1RkEvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=. We may want to express this rate [now denoted LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRK1syNTUsMCw1MV1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk9GPA==]with different discrete compounding frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjA0XUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JVEiMkYnRjUvRjlRJ25vcm1hbEYnRjVGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkE= . Further assume that the rate has duration of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiVEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRidGLy9GNUY5Rjc= years.We solve this problem using non-arbitrage concept: investing at rates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= must yield the same result, otherwise any pricing anomaly will be arbitraged 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 finds a general conversion formula that can be used to obtain an appropriate rate for given input:
Eg: T = 3 Years, r = 5% with annual frequency: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg===1. What is the equivalent rate with a quarterly frequency?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The rate with quarterly compounding is lower [ 4.9088%] - as expected - due to reinvestment effectWhen rates are continuously compounded, the task is not very much different...Now assume that rate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsyNTUsMCwxMDJdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is continuously compounded. What is the equivalent rate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsyNTUsMCwxMDJdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== with a discrete frequency compounding LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjA0XUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JVEiMUYnRjUvRjlRJ25vcm1hbEYnRjVGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjVGQQ== ? 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Expressing discrete rate in terms of continuously compounded rate and vice versa...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Continuously compounded rates will always be lower than rates with discrete compounding due to the compounding effect'
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Discounting</Font></Text-field>Discounting future cash flows:The present value of the a single CF received in the future:
A) Discrete case: 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 = 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
B) Continuously compounded case: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However, what is the present value of a stream of CF for a given period of time (n-years?) - i.e. the PV of Annuity?This quantity depends on the compounding frequency:A): Discrete compounding: 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B): Continuous compounding: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNiQtSSNtbkdGJDYkUSIwRidGMkYyLUYjNiQtSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHRjwvRjNRJ2l0YWxpY0YnRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRmZuLUZTNiVRIkFGJy9GV1EmZmFsc2VGJ0YyLUYvNi1RJyZzZG90O0YnRjIvRjZGXW8vRjlGXW8vRjtGXW8vRj5GXW8vRkBGXW8vRkJGXW8vRkRGXW9GRUZILUklbXN1cEdGJDYlLUYvNi1RLyZFeHBvbmVudGlhbEU7RidGMkY1RjgvRjtGN0Y9L0ZARjdGQUZDRkUvRklRLDAuMTExMTExMWVtRictRiM2JS1GLzYtUSomdW1pbnVzMDtGJ0YyRmFvRmJvRmNvRmRvRmVvRmZvRmdvL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRmhwLUZTNiVRI3JzRidGVkZYRjJGWi1GLzYtUSJ+RidGMkZhb0Zib0Zjb0Zkb0Zlb0Zmb0Znb0ZFRkgtRi82LVEwJkRpZmZlcmVudGlhbEQ7RidGMkY1RjhGXnBGPUZfcEZBRkNGRUZILUZTNiVRInNGJ0ZWRlhGMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 of \302\2431 received over the period of 5Y semi-annually: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PV of \302\2431 received over the period of 5Y continuously: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The annuity factor for continuously compounded case is higher, again due to higher compounding frequency
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Fixed Income Mathematics</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Bonds</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Fixed Coupon</Font></Text-field>Let's define fixed-coupon bond as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEiQkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2ZvcmVncm91bmRHUSpbMCwwLDIwNF1GJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjhRJ25vcm1hbEYn with coupon LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEiQ0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2ZvcmVncm91bmRHUSpbMCwwLDE1M11GJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjhRJ25vcm1hbEYn, yield-to-maturity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEieUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2ZvcmVncm91bmRHUSpbMCwwLDE1M11GJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjhRJ25vcm1hbEYn , annual payment frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEibUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2ZvcmVncm91bmRHUSpbMCwwLDE1M11GJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUSJ+RidGLy9GOEY8RjovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGU0YvRkFGOg==and maturity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEibkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2ZvcmVncm91bmRHUSpbMCwwLDIwNF1GJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjhRJ25vcm1hbEYn years. The value of the bond can be expressed as a present value of future cash flows: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What is the coupon of the bond that gives me the Par Value (100) ?
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How risky is the Bond - i.e. what is the Bond's sensitivity to the underlying yield?
The first-order risk is called Duration, while the second-order effect is known as Convexity: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Maple easily finds a symbolic form of both bond's derivatives and these can be evaluated with arbitrary set of given parameters:
Eg: 10Y bond with 5% coupon payable semi-annually [m=2] and yield to maturity 5%: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Duration of a bond is always negative, while Convexity will always be positive.Duration of a bond decreases as bond's yield increases\342\207\222bond will "repay" quicker when yield is high.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
<Text-field style="Heading 3" family="Arial" layout="Heading 3"><Font family="Arial">Floating rate note<Font italic="false"> [FRN]</Font></Font></Text-field>FLOATING RATE NOTE - a bond with coupon that fluctuates with market movement for short-term rates (typically 3M or 6M money-market rate such as Libor)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In the FRN formula, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiTEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lK2ZvcmVncm91bmRHUSpbMCwwLDIwNF1GJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1GLzYmUSJpRidGNUY3L0Y7USdpdGFsaWNGJ0Y3L0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkc= is the i-th period short-term interest rate (Libor), m represents a coupon margin and s is a credit spread => factor reflecting credit riskiness of an issuer.By how much the FRN price changes if the credit-worthiness of the issuer deteriorates? We are interested in FRN's sensitivity with respect to the credit spread s
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Maple is capable of handling this complex dependency with ease...JSFH
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">CMT</Font></Text-field>Constant-maturity Treasury is a bond with variable coupon where the coupon value is linked to a longer-dated yield on other (typically) government securities. The problem in evaluation of CMTs is reduced to the determination of future coupons: (i) forward par rates and (ii) convexity correction.Let's define forward par rate as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and convexity term as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=Convexity is further defined as second-order Taylor expansion w.r.t yield and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInlGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is used to express a non-linear relationship between bond's price and yield.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CMT floater is then defined as FRN with coupons being equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImlGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI6RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURj0=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's powerful symbolic engine returns the desired formula and is capable of finding any desired comparative static of this instrument. We can differentiate the CMT formula to obtain the first and second derivative of the CMT price w.r.t to yield: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Despite complexity, Maple finds the solution easily
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Asset Swap Package on CMT</Font></Text-field>CMTs are frequently compared or swapped into a standard FRN. Investors may want to swap a longer-dated CMT par yield for a short-term money-market yield when they expect the shorter rates to outperform the longer-dated yields. In CMT Asset Swap package, one needs to determine the value of a spread over money-market rate (say: 3M Libor) that makes the both legs of transaction equal:The asset swap on CMT is then expressed as as Asset Swap Spread (AS)Floating leg definition: L[i] = i-th Libor rate, s = spread, N = notional principal, DF[i] = i-th Discount factor, \316\264 = year 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CMT Leg definition: CMT[i] = i-th CMT coupon, \316\276 = CMT spread, DP = "dirty price of a bond", Par = Par 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
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Derivative Products</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Standard Financial Options</Font></Text-field>What is financial option?Contract that gives the right (but not obligation) to enter into a certain financial transaction: a purchase or sale of financial security - stock, bond etc.There are two basic types of options: CALL \342\207\222Right to buy PUT \342\207\222 Right to sellIn mathematical notation an option is a discounted expectation of future payoff: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZiby1JI21pR0YkNiNRIUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjQvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictSSNtb0dGJDYvUSJ+RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0ZGLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZLLyUpc3RyZXRjaHlHRksvJSpzeW1tZXRyaWNHRksvJShsYXJnZW9wR0ZLLyUubW92YWJsZWxpbWl0c0dGSy8lJ2FjY2VudEdGSy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlpGPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj1GPUY9Rj0tRiw2J1ElQ2FsbEYnRkEvJSdpdGFsaWNHRktGREZHRj0tRj42L1EiPUYnRkFGREZHRklGTEZORlBGUkZURlYvRllRLDAuMjc3Nzc3OGVtRicvRmZuRmBvRj0tRiw2J1EkZXhwRidGQUZqbkZERkctSShtZmVuY2VkR0YkNiYtRiM2KS1GPjYvUSomdW1pbnVzMDtGJ0ZBRkRGR0ZJRkxGTkZQRlJGVEZWL0ZZUSwwLjIyMjIyMjJlbUYnL0ZmbkZecC1GLDYnUSJyRidGQS9GW29GQy9GRVEsYm9sZC1pdGFsaWNGJ0ZHLUY+Ni9RJyZzZG90O0YnRkFGREZHRklGTEZORlBGUkZURlZGWEZlbi1GLDYnUSJURidGQUZjcEZkcEZHRkFGREZHRkFGREZHRmZwLUYsNidRKSYjNTg2ODQ7RidGQUZjcEZkcEZHLUY+NjBGQEZBRmNwRmRwRkdGSUZMRk5GUEZSRlRGVkZYRmVuLUY+NjBRIltGJ0ZBRmNwRmRwRkcvRkpGQ0ZML0ZPRkNGUEZSRlRGVi9GWVEsMC4xNjY2NjY3ZW1GJy9GZm5GZ3EtRiw2J1EkTWF4RidGQUZjcEZkcEZHLUY+Ni9RIihGJ0ZBRkRGR0ZkcUZMRmVxRlBGUkZURlZGZnFGaHEtRiw2J1EiU0YnRkFGY3BGZHBGRy1GZm82KC1GIzYmRmlwRkFGREZHRkFGREZHLyUlb3BlbkdGY3EvJSZjbG9zZUdRIl1GJy1GPjYvUSgmbWludXM7RidGQUZERkdGSUZMRk5GUEZSRlRGVkZdcEZfcC1GLDYnUSJLRidGQUZjcEZkcEZHRkFGREZH,0)]
Put = 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,0)]
So, to value an option we need to define the stochastic path for the underlying security: Following Black-Scholes-Merton continuous-time model we assume that the underlying security (say stock price) follows a geometric Brownian motion: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here r is constant drift coefficient and \317\203 is constant (positive) volatility coefficientSince the terminal value of this process is well-known, using Ito formula we can derive the model's value explicitly: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y is a standard normal variable: 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 the Standard Normal 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Determine the critical value for which the stock's terminal value is equal to the strike price K: S[T] = 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Specify the option's payoff: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 an option: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This is the famous Black-Scholes option formula for a stock price that pays no dividends in "non-abbreviated form" \342\207\222functional parameters 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are included in the pricing formula.We can substitute actual values to obtain numerical form of the expression and plot it against two fundamental variables: stock price LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== and volatility \317\203: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It is worth mentioning that the put value increases when the volatility gets higher even if the option's intrinsic value max(K-S[T],0) is zero .
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Option's sensitivities</Font></Text-field>Valuing option is only one part of the story....We also need to find the ways how to hedge an option and understand option's sensitivities w.r.t functional parameters. This is the area where Maple comes particularly handy...We define four basic option sensitivities (using Call Option):a) Option Delta \342\206\222 first derivative of option price w.r.t stock price
b) Option Gamma \342\206\222 second derivative of option price w.r.t stock price
c) Option Vega \342\206\222 first derivative of option price w.r.t volatility
d) Option Theta \342\206\222 first derivative of option price w.r.t time (time decay)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We can visualize the sensitivities in any desired format....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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Non-Standard Options</Font></Text-field>Black-Scholes option pricing formula is a standard option with piecewise linear payoff. However, we can use Maple's symbolic engine to derive various non-standard option formulas where the terminal payoff is not any longer piecewise linear.
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Power Option</Font></Text-field>The option payoff is now the square of the stock price: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
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Powered Option</Font></Text-field>The payoff is the difference between the stock price and the strike price squared: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
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Digital</Font></Text-field>Option will pay a pre-determined amount PA if S[T] \342\211\245 KLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbWlHRiQ2JVEoRGlnaU9wdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJGV4cEYnL0YwRj1GOS1JKG1mZW5jZWRHRiQ2JC1GIzYoLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZmbi1GLDYlUSJyRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZfby1GLDYlUSJURidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjlGW28tRiw2JVEkaW50RidGL0YyLUZUNiQtRiM2Li1GLDYlUSNQQUYnRi9GMkZbby1GLDYlUSVucGRmRidGL0YyLUZUNiQtRiM2JS1GLDYlUSJ5RidGL0YyRmRvRjlGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRl5vL0ZOUSwwLjMzMzMzMzNlbUYnRmdwLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUklbXN1YkdGJDYlLUYsNiVRJWNyaXRGJ0YvRjItRiM2JS1GLDYlUSR2YWxGJ0YvRjJGZG9GOS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIRmVuRmBvLUYsNiVRKWluZmluaXR5RidGL0YyRmRvRjlGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRl5vRmBvLUY2Ni1RKWFzc3VtaW5nRidGOUY7Rj5GQEZCRkRGRkZIRl5vRmBvRmdyLUYsNiVRJnNpZ21hRidGUkY5LUY2Ni1RIj5GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiMEYnRjlGanBGaG5GYHNGY3NGanBGYW9GYHNGY3MtRjY2LVEiO0YnRjlGO0ZdcUZARkJGREZGRkhGXm9GTUZkb0Y5
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Range</Font></Text-field>Option that pays a fixed amount A if the stock price at option maturity is within the range of [LP,HP]: 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
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Performance Option</Font></Text-field>The option will pay amount A if the stock price at at time T exceeds its value at time U S[T] > S[U] with U< 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We need to determine the critical value for the option 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We can now value the option: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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Alternative Process Dynamics</Font></Text-field>BS process is a representative example of multiplicative geometric Brownian motion. We can define alternative process - the general stochastic process - using Bachelier normal dynamics - in our particular example as driftless: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 need this process to model securities paths where value is allowed to become negativeCMT coupons (mentioned earlier) are frequently issued with a Zero-Floors V = \356\224\274[Max(CMT,0)]We can use Bachelier process to value this embedded optionality directly: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Obtaining the sensitivity is 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<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">PDEs in Finance</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Finding probability density function [PDF] for B/S process</Font></Text-field>SDE with the initial condition determines a diffusion process. In finance SDEs are used in two fundamental ways: (i) defining an expected payoff, and (ii) finding transitional probability at being in a certain state at a given timeThese views are different, but not unrelated \342\207\222BS PDE solves the backward Kolmogorov PDE, whereas probability density [PDF} solves the forward Kolmogorov equation - being itself an adjoint of the backward PDESDE process for a financial product: 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 where a = drift and b = diffusion coefficientThe process can be expressed in terms of diffusion 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 solve this PDE analytically, we will try to change it into constant-coefficient PDE by change of 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 want to solve this equation with initial condition 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 the PDE is now in "desirable format", we apply Fourier transform to solve it:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2JVEnYXNzdW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNi8tRiw2JVEmc2lnbWFGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSI+RidGPy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlUtSSNtbkdGJDYkUSIwRidGPy1GQjYtUSIsRidGP0ZFL0ZIRjFGSUZLRk1GT0ZRL0ZUUSYwLjBlbUYnL0ZXUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZYRmZuLUYsNiVRInJGJ0YvRjJGQUZYLyUrZXhlY3V0YWJsZUdGPkY/Rj8tRkI2LVEiOkYnRj9GRUZHRklGS0ZNRk9GUUZTRlYtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGXnAvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRmpvNiZGXHBGX3BGYnAvRmVwUSVhdXRvRictRiw2JVEqaW50ZXJmYWNlRidGL0YyLUY2NiQtRiM2Jy1GLDYlUSxzaG93YXNzdW1lZEYnRi9GMi1GQjYtUSI9RidGP0ZFRkdGSUZLRk1GT0ZRRlNGVkZYRmRvRj9GP0Zmb0ZncEZpby1GLDYlUShmb3VyaWVyRidGL0YyLUY2NiQtRiM2Ky1GLDYlUSRwZGVGJ0YvRjJGZm4tRkI2LVEifkYnRj9GRUZHRklGS0ZNRk9GUUZqbi9GV0Zbby1GLDYlUSJ4RidGL0YyRmZuRmJyLUYsNiVRIndGJ0YvRjJGZG9GP0Y/LUZCNi1RIjtGJ0Y/RkVGaW5GSUZLRk1GT0ZRRmpuRlZGaW8tRiw2JVElc3Vic0YnRi9GMi1GNjYkLUYjNitGaHEtRjY2JC1GIzYqLUYsNiVRInVGJ0YvRjItRjY2JC1GIzYnRmZyRmZuRl5vRmRvRj9GP0ZmbkZmckZmbkZpckZkb0Y/Rj9GZXEtRiw2JVEidkYnRi9GMi1GNjYkLUYjNiVGXm9GZG9GP0Y/RmZuLUYsNiVRIiVGJ0YvRjJGZG9GP0Y/RlxzRmRvRj8=We have now removed a spatial variable from the equation \342\207\222 transformed PDE into ODE, which we can now solve with the given initial condition: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Since Fourier transform in invertible, we retrieve back the desired probability density function: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Finally, we want to check the correctness of our result: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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">PDF in higher dimension</Font></Text-field>Using suitable change of variables, Maple can efficiently solve PDEs in higher dimensions using Fourier 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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">PDF for Jump-Diffusion</Font></Text-field>Standard B/S process assumes constant volatility and normal distribution of log-returnsIn reality, the empirical distribution is not normal \342\207\222returns exhibit "fat tails": extreme values probabilities are higher than the one suggested by normal distributionJump-diffusion is one of the solutions that addresses the non-normality of the underlying processHow does this work? Standard diffusion is "enriched" with jumps 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Jump process: - Poisson process with intensity \316\273 \342\207\222 jump is distributed with Poisson PMF, and over the given time period there are in total N jumps with magnitude JPDE for jump-diffusion: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5ODI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIkpGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== is the PDF of the jumpAgain, specify initial condition: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Apply Fourier transform to the 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 knows how to apply Fourier transform to convolution, and converts spatial variable into parameter. We have obtained ODE that we will solve with the given initial condition: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We will try to invert it back to retrieve the original 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Maple successfully finds a "closed-form" solution, which can be further simplified by specification of the jump density
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Numerical Computation</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Monte Carlo method for standard option</Font></Text-field>Here we demonstrate how Maple's Statistics package can be efficiently used to value European-type financial options numericallyWe need to define stock price process and simulation parameters:
S[0] = 20; \317\203=25%; time = 1Y, time bucket = T/N = 1/100; number of stock price simulations = 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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Monte Carlo method with stochastic volatility</Font></Text-field>We now allow the volatility of a financial product to follow its own stochastic process - correlated with the underlying security:We assume that the volatility of the stock \317\203 follows driftless Brownian motion with constant volatility parameter \316\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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Normal option with stochastic volatility - numerical integration</Font></Text-field>Here we show how Maple's numerical integration can be efficiently applied to solve the option pricing when the volatility itself is stochasticWe define the underlying price process as Bachelier normal process with drift \316\274 and stochastic volatility \317\203[T]We also assume that the volatility follows a geometric Brownian motion with drift \316\273 and constant diffusion parameter \316\267Two Wiener process LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjVzFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1cyRidGL0YyRjk= are correlated with coefficient \317\201\342\207\222 the underlying density is bi-variate 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Determine the critical value for the underlying process: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Write down the option formula - example using Put Option: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 the Put option numerically, using Maple's numerical integration techniques: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 we derived the option price numerically, we can still use Maple to calculate for us option sensitivities, for example 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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Numerical PDE method</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Standard Black-Scholes approach</Font></Text-field>Define the drift and diffusion parameters of BS PDE: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Specify the numerical values and Initial / Boundary conditions
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 the PDE 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Visualize the result:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVElc29sbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzotRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEncGxvdDNkRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNjctRiw2JVEkdGF1RicvRjBGPUY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiMEYnRjktRjY2LVEjLi5GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5RJjAuMGVtRictRmluNiRRIjFGJ0Y5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktGYm8vRk5RLDAuMzMzMzMzM2VtRictRiw2JVEiU0YnRlpGOUZlbi1GaW42JFElMC4wMUYnRjlGXG8tRmluNiRRJDAuMUYnRjlGZm8tRiw2JVElYXhlc0YnRlpGOUZlbi1GLDYlUSZib3hlZEYnRlpGOUZmby1GLDYlUSZ0aXRsZUYnRlpGOUZlbi1GUzYmLUYjNiotRiw2I1EhRictSSNtc0dGJDYjUTZDYWxsfk9wdGlvbn5Fdm9sdXRpb25GJ0Zmby1GLDYlUSVmb250RidGL0YyRmVuLUZTNiYtRiM2Jy1GLDYlUSVCT0xERidGL0YyRmZvLUZpbjYkUSMxNEYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZnckY5RjlGaXJGXHNGZ3JGOUY5LUY2Ni1RIjtGJ0Y5RjtGaW9GQEZCRkRGRkZIRmpvRk0tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGZ3MvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZqb0Zhby1GLDYlUSRyZXNGJ0YvRjItRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUYrLUY2Ni1GOEY5RjtGPkZARkJGREZGRkhGam9GYW8tRiw2JVEmdmFsdWVGJ0YvRjItRlM2JC1GIzYlRmNxRmdyRjlGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZic0ZgdEZjdC1GUzYkLUYjNictRmluNiRRJDAuNUYnRjlGZm8tRmluNiRRJTAuMDVGJ0Y5RmdyRjlGOUZfc0ZnckY5
<Text-field style="Heading 3" layout="Heading 3"><Font family="Arial">Alternative PDE process</Font></Text-field>JSFHLet's examine the square-root process for security S: 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parameters values and IBCs and solve PDELUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZQLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEmMC4wMTNGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GWi1GLDYlUSJHRidGL0YyRjUtRlA2JFEmMC4wNzVGJ0Y5RlMtRiw2JVEoJnNpZ21hO0YnRi9GMkY1LUZQNiRRJTAuMjVGJ0Y5RlNGVi1GLDYlUSJNRidGL0YyRjUtRlA2JFEkMC4zRidGOUZTLUYsNiVRIkxGJ0YvRjJGNS1GUDYkUSIwRidGOUZTRlYtRiw2JVEickYnRi9GMkY1LUZQNiRRJTAuMDRGJ0Y5RlMtRiw2JVEiVEYnRi9GMkY1LUZQNiRRIjFGJ0Y5RlMtRiw2JVEiS0YnRi9GMkY1LUZQNiRRJjAuMDY1RidGOUZTRmRwRjVGZ3BGUy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0ZlcS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUSRpYmNGJ0YvRjItRjY2LVEjOj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkobWZlbmNlZEdGJDYmLUYjNjctRiw2JVEiRkYnRi9GMi1GZXI2JC1GIzYnLUYsNiVGZnAvRjBGPUY5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkhGWS9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJTRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEqcGllY2V3aXNlRidGL0YyLUZlcjYkLUYjNitGaXMtRjY2LVEvJkdyZWF0ZXJFcXVhbDtGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRmpwRmNzRmlzLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GX3VGanBGXHRGOUY5RmNzLUYsNiVRIkRGJ0Zic0Y5LUZlcjYmLUYjNiUtRlA2JFEiMkYnRjlGXHRGOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZlcjYkLUYjNiZGVkZpckZcdEY5RjktRmVyNiQtRiM2Jy1GLDYlUSYmdGF1O0YnRmJzRjlGY3NGYm9GXHRGOUY5Rl50RmdwRmNzRmF1RmR1RmF2LUZlcjYkLUYjNidGaXZGY3NGaG9GXHRGOUY5Rl50RltwRlx0RjlGOS9GXHZRInxmckYnL0ZfdlEifGhyRictRjY2LVEiO0YnRjlGO0Zmc0ZARkJGREZGRkhGWUZNRlx0Rjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVElc29sbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEocGRzb2x2ZUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzZCLUYsNiVRJHBkZUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJGliY0YnRi9GMkZaLUYsNiVRKG51bWVyaWNGJ0YvRjJGWi1GLDYlUSV0aW1lRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRJyYjOTY0O0YnL0YwRj1GOUZaLUYsNiVRKXRpbWVzdGVwRidGL0YyRmVvLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiL0YnRjlGO0Y+L0ZBRjFGQkZERkZGSC9GS1EsMC4xNjY2NjY3ZW1GJy9GTkZocC1GYHA2JFEkMTAwRidGOUZaLUYsNiVRKnNwYWNlc3RlcEYnRi9GMkZlb0ZfcEZjcC1GYHA2JFEkMjUwRidGOUZaLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGaG4vRk5GaW4tRiw2JVEnbWV0aG9kRidGL0YyRmVvLUYsNiVRL0NyYW5rTmljaG9sc29uRidGL0YyRlotRiw2JVEpb3B0aW1pemVGJ0YvRjJGZW8tRiw2JUYxRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUY2Ni1RIjtGJ0Y5RjtGZ25GQEZCRkRGRkZIRmhuRk1GYnJGOQ==Visualize the 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Note the difference in option profile at option maturity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyLUY2Ni1RIn5GJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktGVEYy[time value = 0], and on inception \317\204 = 0:: \342\206\222 full benefits of option time value.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Data Analysis</Font></Text-field>Maple has a broad range of data fitting routines to analyze time seriesThis particular example focuses on relationship between real and nominal rates in a hypothetical 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We use Maple's Statistics package to test various regression models on the given set of dataModel 1: Logarithmic regression: 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Model 2: User-defined regression: 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Model 3: Polynomial model of 4-th order: 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 of regression models is arbitrary and many other model combinations are feasible.Maple finds least-square parameters of given models easily...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How good is our fit?Maple can calculates the required "goodness-of-fit" statistics for us....
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<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Risk Application</Font></Text-field>In this section we examine the calculation of Value-At-Risk statistics for a small portfolio of financial instruments.Value-at-Risk [Value@Risk] determines how much an institution may loose over a given time (typically number of days), at pre-determined confidence level.
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Practical Approach</Font></Text-field>Assume a portfolio of 5 financial instruments with volatility vector U, weight vector W and correlation matrix 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Determine the size of the portfolio and compute Variance-Covariance Matrix: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Apply weights to obtain portfolio volatility: W' *VarCov * WLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEkVmFyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEjV1RGJ0YvRjItRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRI0NURidGL0YyRlItRiw2JVEiV0YnRi9GMi1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRlVGTS8lK2V4ZWN1dGFibGVHRj1GOQ==This is the portfolio's volatility given weights and correlations shown aboveLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We want to test the Value@Risk in non-standard fashion - by applying "fat-tail" distribution to the portfolio risk. To estimate this impact, we will compare a tail behavior of Normal and Extreme-Value [Gumbel] 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As shown on the chart above, the Gumbel density exhibits "longer" and "fatter" tail in the extreme region. This may be particularly attractive if we want to obtain Value@Risk under alternative distributional measure in non-Normal environment.To obtain the Value@Risk measure, we will need a quantile value of a particular distribution to feed into the Value@Risk formula:
Value@Risk = Market Exposure*Volatility*Quantile*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 see that the quantile of Gumbel density at 99% confidence level is almost double of that of standard normal density:Example: Market exposure = 100 million, t= 10 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The difference in the risk measure is significant: Under extreme-value assumption, the VaR figure is higher by 727,000 than the measure suggested by the standard market approach
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Theoretical Approach</Font></Text-field>How can VaR theory enhance our Value@Risk calculation?Apart from standard VaR measure, we may be interested in "Incremental" VaR - i.e. changes in VaR measure due to variability of inputIt is quite legitimate to ask a question: "By how much my VaR is going to change if a security's volatility, weight or correlation changes?"Define 3-security portfolio with weight vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjdnZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUSxbMjU1LDAsMTUzXUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== and covariance matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlElY21hdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsyNTUsMCwyMDRdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw==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 determined the theoretical portfolio's volatility, we can obtain comparative static for the VaR behavior:By how much will VaR change if volatility of the Asset[1] changes?By how much will VaR change if weight of Asset[1] changes?How VaR will be affected if correlation between Asset[1] and Asset[2] changes?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2JVElZHZvbEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJWRpZmZGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSh0aGVvdmFyRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEmc2lnbWFGJy9GMEY9RjktRlM2Ji1GIzYlLUkjbW5HRiQ2JFEiMUYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zob0Y5RjktRjY2LVEiO0YnRjlGO0ZnbkZARkJGREZGRkhGaG5GTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0ZocC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUShkd2VpZ2h0RidGL0YyRjVGTy1GUzYkLUYjNihGV0ZaLUYsNiVRInZGJ0YvRjJGYG9GaG9GOUY5RmBwRmNwLUYsNiVRJWRyaG9GJ0Zfb0Y5RjVGTy1GUzYkLUYjNihGV0ZaLUYsNiVRJHJob0YnRl9vRjktRlM2Ji1GIzYnRmRvRlotRmVvNiRRIjJGJ0Y5RmhvRjlGOUZqb0ZdcEZob0Y5RjlGYHBGaG9GOQ==
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Conclusion</Font></Text-field>The purpose of this document is to show that Maple is a versatile tool that meets various needs of finance professionals in arbitrary level of complexity. The selection of problems demonstrates just small universe of potential applications of the product in the real-life environment, however actual usage in the finance industry is virtually unlimited. Maple's symbolic manipulation and powerful numerical computation combined with excellent visualization tools make the product an efficient and robust tool for model design, test and validation, however other areas can be accommodated within this framework in equally effective manner.Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended to demonstrate the use of Maple to solve particular set of problems.It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft and/or author. 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