Pricing Models for Valuation of Foreign Exchange Derivatives

Pricing Models for Valuation of Foreign Exchange Derivatives Prepared and presented by Anshum Bhambri, CFE graduate 1

Options Basics 2

Options Definitions Option Buyer has the right but not the obligation to o to buy [Call] or sell [Put] a specified amount of currency o at a pre-agreed price [Strike Price or Exercise Price] o Either during [American] or at the end [European] of a specified period [Maturity or Expiry Date] Option Buyer o will only exercise option if beneficial at expiry (i.e. in the money) o pays a premium for the option to the seller, usually upfront Option Seller o after receipt of premium: contingent liability but no claim 3

Options Payoff Suppose you need to buy EUR / sell USD for delivery in 2 months time o Just buy EUR today at the forward rate and take delivery on the forward date OR o Buy the option to deliver USD against EUR in 2 months time What is the difference? o Obligation versus Choice o Flexibility, Participation, Asymmetry 4

Options Premium 5

Options Vanilla Option Payoffs European Call : Underlying: S Maturity: T Strike: K Payoff: Max(0, S(T) K) European Call : Underlying: S Maturity: T Strike: K European Payoff: Max(0, K - S(T)) Payoff Call @ 80% 100% Payoff Put @ 80% 100% 80% 80% 60% 60% 40% 40% 20% 20% 0% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% S(T) 0% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% S(T) 6

Options Vanilla Option Combinations Bull / Call Spread: long a call at K1, short a call at K2, K1 < K2 Bear / Put Spread: long a put at K1, short a put at K2, K1 > K2 Payoff Bull Spread @ 100%,120% 30% Payoff Bear Spread @ 80%,100% 30% 20% 20% 10% 10% 0% 40% 60% 80% 100% 120% 140% 160% S(T) 0% 40% 60% 80% 100% 120% 140% 160% S(T) Straddle: long a call at K1, long a put at K2, K1 = K2 Strangle: long a call at K1, long a put at K2, K1 > K2 Payoff Straddle @ 100% 100% Payoff Strangle @ 80%,120% 100% 80% 80% 60% 60% 40% 40% 20% 20% 0% 0% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 200% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 200% S(T) 7 S(T) 7

Options Vanilla Option Comninations Butterfly Spread : long a call at K1 short two calls at K2 long a call at K3 K1 < K2 < K3 CFE School Payoff Butterfly @ 80%,100%,120% 30% 20% 10% 0% 50% 70% 90% 110% 130% 150% S(T) 8 8

Options Call or Put? Put Call Parity... 9

Options Problems with options 10

Options Black-Scholes Insight - Two possible outcomes 11

Options Black-Scholes Insight - Two possible outcomes 12

Options Black-Scholes Insight - Two possible outcomes 13

Options Hedge Ratio 14

Options When probability comes into play... However, the expectation argument allows us to infer probabilities 15

Options...option prices change 16

Options Black-Scholes Insight.. Re-visited 17

Options Black-Scholes Insight Several Possible Outcomes 18

Options Black-Scholes Insight Several Possible Outcomes 19

Options Pricing Terminal Distribution 20

Options Pricing Terminal Distribution Method 21

Options Black-Scholes Assumptions Call Option Price can also be expressed as: C rt 1 N( d ) Se Strike PVfr 2 N( d ) N( d ) 2 d 2 ln Spot Strike d1 d 2 vol ( r ( vol vol t t 2 / 2)) t 22

Options Pricing Moves in Spot Price 23

Options Pricing Moves in Strike 24

Options Pricing Moves in Maturity 25

Options Pricing Moves in Volatility 26

Options Pricing Moves in Interest Rates 27

Options Volatility and/or time? 28

Option Pricing Models Black-Scholes Model 29

Black-Scholes Model Stochastic Differential Equation d S / S = µ dt + σ dz Geometric Returns Drift Volatility Brownian Process The Black Scholes model is a mathematical model of the market for an equity, in which the equity's price is a stochastic process (dz) S, the price of the stock; μ, the drift rate of S; σ, the volatility of the stock; this is the square root of the quadratic variation of the stock's log price process; t is time Assumptions: The equity price follows a Geometric Brownian motion with constant drift and volatility The stock does not pay a dividend There are no transaction costs (i.e. markets are frictionless) Trading may take place continuously There is no prohibition on short selling The risk free rate is the same for borrowing and lending Assets are perfectly divisible There is no arbitrage opportunity 30

Black-Scholes Model Closed-form Solution for ATM Vanilla Options Using No Arbitrage theory, a stock Forward is Calculated as: F S 0 e r f d repo T Here r f is the risk-free rate, d is the dividend yield, and repo is the rate of stock borrowing. Call option price can be expressed in terms of Forward price F : In other words, Call Option Price = PVf x { Forward x Delta Strike x Moneyness } 31

Black-Scholes Model Closed-form Solution for ATM Vanilla Options Call Option Price can also be expressed as: C N( d ) N( d ) rt 1 N( d ) Se Strike PVfr 2 2 d 2 ln Spot Strike d1 d 2 vol ( r ( vol vol t t 2 / 2)) t In other words, Call Option Price = { Probability of Exercise x (Expected Value if Exercised Strike) } x PVf The price of a Put Option can be given as: r T t t Ke N d SN d where, N d N d 1 P S, 2 1 1 1 32

Black-Scholes Model Model Limitations What it can price: ATM Vanilla Options Flow traders back out market implied volatilities for vanilla options given strike level and maturity. This helps build the volatility surface What it cannot price: Call Spreads/Digitals Volatility input in Black-Scholes Model is assumed to be independent of Spot. Thus, the model cannot price Skew 33

Black-Scholes Model Model Parameters Various parameters that go into the model are: T is the time to maturity User Input S is the spot price of the underlying asset Market Data (Reuters) K is the strike price User Input r is the risk free rate Market Data (Reuters) σ is the volatility in the log-returns Trader Input (Vol surface from traded options) 34

Black-Scholes Model Monte Carlo Simulation The Monte Carlo option pricing methodology involves simulating price patterns for the underlying asset The first task entails creating a series of random numbers on which the prices will eventually be based For each set of random numbers an asset price path is calculated and the payoff of the option corresponding to this path is calculated this represents a single simulation The process is repeated a large number of times and an estimate for the expected value of the payoff is obtained by taking the average of the answers from all the simulations We then calculate the present value of this expected payoff to get the option value 35

Black-Scholes Model Going beyond Black-Scholes Price of a structure/option is the expected value of returns at maturity We can price using Monte Carlo whereby we simulate several paths for each underlying and calculate price as the expected value of returns over these paths Expectation is the sum over a product of Payoff: Given a path of the underlying, payoff is determined by the formula description Probability of occurrence: Every simulated path occurs with a certain probability Probability of occurrence is a function of the diffusion processes of the underlying Stock process: The drift and volatility assumptions Volatility process(if not Black Scholes): The model parameters, typically reversion speed, long-term mean and vol-of-vol To simulate Monte Carlo mentally, think of Possible scenarios: eg, if there are barriers, how many end-states can there be for the underlying Boundary conditions: eg, in a basket, what happens if correlation goes to 0, or 1 or -1 Objective of the exercise is to Check feasibility of price by determining logical bounds Get an intuitive understanding of risks involved 36

Option Greeks Delta, Gamma, Theta, Vega, Rho 37

The Greeks Introduction The value of an option depends upon Underlying Price (Delta, Gamma) Time to Expiration (Theta) Volatility (Vega) Interest Rates (Rho) Changes in these variables will change the theoretical value of an option. The Greeks are the quantities representing the sensitivities of options to a change in one of the underlying parameters on which the value of the underlying asset depends Collectively these have also been called the risk sensitivities, risk measures or hedge parameters 38

Delta Introduction CFE School The Delta (Δ) measures the rate of change of option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value, V, of a portfolio of derivative securities on a single underlying instrument, S, with respect to the underlying instrument's price. Delta can be treated as a Forward Equivalent. Buying the call is like buying delta times the forward and buying a put is like selling delta times the forward. Given a call and a put option for the same underlying, strike price and time to maturity, The sum of the absolute values of the delta of each option will be 1 (can be deduced from Put-Call Parity) Delta v/s Moneyness: Deltas range from 0.00 for deep out-of-the-money calls to 1.00 for deep in-the-money calls and from 0.00 for deep out-of-the-money puts to -1.00 for deep in-the-money puts Approximately, at-the-money calls have deltas of 0.50 and at-the-money puts have deltas of -0.50 39

Delta Delta CFE School Delta Sensitivity to Volatility and Time to Expiry The delta of an ATM option does not change much with a change in volatility or time to expiration. Intuitively, this makes sense, as the likelihood that an ATM option will end up in the money is roughly the same as the likelihood that it will end up out of the money An increase in volatility or time to expiration increases the likelihood that an OTM option ends up in-themoney at maturity. Hence an increase (decrease) in volatility or time to expiration increases (decreases) the delta of an OTM option Conversely, an increase in volatility or time to expiration decreases the delta of an ITM Option European Call - Delta profile European Put - Delta profile 100% 10% 90% 80% 12m 11m 10m -10% 12m 11m 10m 70% 60% 50% 40% 30% 9m 8m 7m 6m 5m 4m -30% -50% -70% 9m 8m 7m 6m 5m 4m 20% 10% 3m 2m 1m -90% 3m 2m 1m 0% -110% 55% 65% 75% 85% 95% 105% 115% 125% 135% 145% 55% 65% 75% 85% 95% 105% 115% 125% 135% 145% Underlying level Underlying level 40

Gamma Introduction CFE School The gamma ( ), sometimes referred to as the curvature of the option, is the rate at which an option s delta changes as the price of the underlying changes The gamma is the second derivative of the value function with respect to the underlying price The gamma is a measure of how fast an option changes its directional characteristics and is thus a measure of directional risk a large gamma number indicates a high degree of directional risk and vice versa As a tool, gamma can tell you how "stable" your delta is. A large gamma means that your delta can start changing dramatically for even a small move in the stock price Gamma is always positive and equal in value for vanilla calls and puts. It is given as: Gamma is a benefit when you have a long option position and it is a cost when you have a short option position i.e. being long an option is long gamma and vice versa 41

Gamma CFE School Gamma Sensitivity to Price of the Underlying ATM Options have higher gamma than both ITM and OTM Options with otherwise identical contract specifications. This means that the delta of ATM options changes the most when the stock price moves up or down The gamma is maximum when the strike price of the option is: Intuitively, this is when the strike price of the option is slightly greater than the forward price European Put - Gamma profile 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 55% 65% 75% 85% 95% 105% 115% 125% 135% 145% 12m 11m 10m 9m 8m 7m 6m 5m 4m 3m 2m 1m Underlying level 42

Gamma Sensitivity to Volatility and Time to Expiry Gamma v/s Volatility: For ATM options, if we decrease (increase) our volatility assumption, the option increases (decreases) with otherwise identical contract specifications. gamma of the For ITM/OTM options, if we increase (decrease) our volatility assumption, the gamma of the option increases (decreases) with otherwise identical contract specifications. Gamma v/s Time to Expiration: For ATM options, the gamma increases as expiration approaches. The gamma of a short-term ATM option will always be greater than the gamma of a longer-term ATM option with otherwise identical contract specifications. For ITM/OTM options, the gamma of a short-term option will always be lesser than the gamma of a longer-term option with otherwise identical contract specifications. 43

Theta Introduction The theta ( ) or the time decay factor, is the rate at which an option loses value as time passes and there is no movement in either the stock price or volatility. i.e. all other conditions remain the same. It is usually expressed in points lost per day A long option position loses value as time passes and hence it will always have a negative theta whereas a short option position will always have a positive theta Theta v/s Moneyness: ATM options have greater thetas than either ITM or OTM options with otherwise identical contract specifications. This makes sense as ATM options have the highest extrinsic value, so they have more extrinsic value to lose over time than an ITM or OTM option Theta v/s Volatility: As we decrease (increase) our volatility assumption, the theta of an option will increase (decrease) Theta v/s Time to Expiration: For ATM options, the theta increases as expiration approaches. The theta of a short-term ATM option will always be greater than the theta of a longer-term ATM option with otherwise identical contract specifications For ITM/OTM options, the theta of a short-term option will always be lesser than the theta of a longer-term option with otherwise identical contract specifications 44

Theta Relationship with Gamma As a general principle, an option position will have a gamma and a theta of opposite signs. Moreover, the relative size of the gamma and theta will also correlate. A large positive gamma goes hand in hand with a large negative theta and vice versa Every option position is thus a trade off between market movement and time decay. If price movement in the underlying contract will help the trader (positive gamma), the passage of time will hurt (negative theta) and vice versa. The trader cannot have it both ways. He either wants the market to move, or he wants it to sit still Just as a large gamma represents a high degree of risk with respect to market movement, a large theta represents a high degree of risk with respect to passage of time Question: Using what you have learnt so far, construct a portfolio that is long theta & long gamma 45

Vega CFE School Vega Introduction The vega ( ) is an estimate of how much the theoretical value of an option changes when volatility changes by 1% Higher volatility means higher option prices. The reason for this is that higher volatility means a greater price swing in the stock price, which translates into a greater likelihood for an option to make money by expiration Vega is the derivative of the option value with respect to the volatility of the underlying. Since all options gain value with rising volatility, the vega for both calls and puts is positive. Vega v/s Moneyness: ATM options have greater vegas than either ITM or OTM options with otherwise identical contract specifications. This means that the value of ATM options changes the most when the volatility changes As a corollary, the OTM option is always the most sensitive in percent terms, to a change in volatility European Put - Vega profile 14% 12m 12% 11m 10m 10% 9m 8m 8% 7m 6% 6m 5m 4% 4m 3m 2% 2m 1m 0% 50% 70% 90% 110% 130% 150% 170% Underlying level 46

Vega Sensitivity to Volatility and Time to Expiry Vega v/s Volatility: For an ATM option, the vega is relatively constant with respect to changes in volatility. For ITM/OTM options, if we increase (decrease) our volatility assumption, the vega of the option increases (decreases) with otherwise identical contract specifications. The sensitivity of the vega to volatility is known as the Volga or the Volgamma or the Vol-of-Vol. Vega v/s Time to Expiration: The vega of an option decreases as the time to expiration grows shorter. i.e. A longer-term option always has a higher vega than a shorter-term option with otherwise identical contract specifications 47

Rho Introduction The Rho ( ) is an estimate of how much the theoretical value of an option changes when interest rates move by 1% In case of a stock option, a call option will have a positive rho as an increase in interest rates will make buying a call option a more desirable alternative to borrowing money at higher interest rates and buying the stock Conversely, a put option will have a negative rho as an increase in interest rates will make buying a put option a less desirable alternative to selling the stock and investing the proceeds at higher interest rates Rho v/s Moneyness: Deep ITM options have the highest rho (in absolute terms) because they require the greatest cash outlay Rho v/s Time to Expiration: The greater the time to expiration, the higher is the rho Although, changes in interest rates can affect an option s theoretical value, the interest rate is usually the least important of the inputs into a pricing model. So, the rho is not considered as critical as the other sensitivities 48

Summary Sensitivity of an Option Price to Changing Market Conditions 49

Summary Analysis of Greeks of Vanilla Options 50

Structures in FX markets Basics 51

Structures in FX Markets Risk reversal A risk-reversal consists of being long (buying) an out of the money call and being short (i.e. selling) an out of the money put, both with the same maturity. A risk reversal is a position in which you simulate the behavior of a long, therefore it is sometimes called a synthetic long. This is an investment strategy that amounts to both buying and selling out-of-money options simultaneously. In this strategy, the investor will first make a market hunch, if that hunch is bullish he will want to go long. However, instead of going long on the FX pair, he will buy an out of the money call option, and simultaneously sell an out of the money put option. Presumably he will use the money from the sale of the put option to purchase the call option. Then as the FX pair goes up in price, the call option will be worth more, and the put option will be worth less. 52

Structures in FX Markets Risk reversal A Risk Reversal can refer to the manner in which similar OTM Call and put options, usually FX options, are quoted by dealers. Instead of quoting these options' prices, dealers quote their volatility. For eg. R 25 = σ call,25 σ put,25 In other words, for a given maturity, the 25 risk reversal is the vol of the 25 delta call less the vol of the 25 delta put. The 25 delta put is the put whose strike has been chosen such that the delta is -25%. The greater the demand for an options contract, the greater its volatility and its price. A positive risk reversal means the volatility of calls is greater than the volatility of similar puts, which implies a skewed distribution of expected spot returns composed of a relatively large number of small down moves and a relatively small number of large up moves. 53

Structures in FX Markets Range Forward A range forward provides the client with a definite range of FX exposure, with downside protection and potential to sell at a premium to current price If the investor is long the forward on a FX pair, he can enter into a range forward by purchasing a put option at a lower strike (as compared to current spot) and selling a call option at a higher strike. The strikes are chosen such that the effective cost of the structure is zero. Advantages o No upfront premium i.e. Zero Cost o The client has full participation to the upper limit and full protection below the floor level Payoff (%) Cap Downside protection Upside participation (Capped) 0% 0 Put Strike Call Strike Underlying price 54

Structures in FX Markets Interpolating option prices from quotes 55

Structures in FX Markets Calculating other deltas 56

Structures in FX Markets Interpolating the risk reversal 57

Structures in FX Markets Interpolating the strangle 1.3% 58

The Book of Greeks Quantitative Finance Lecture Notes Write to cfeschool@risklatte.com for more info on CFE Course and the CFE Exam. 59